On the spectral determinations of the connected multicone graphs $ K_r\bigtriangledown sK_t $

TL;DR

This paper proves that connected multicone graphs, formed by joining a clique with multiple copies of a complete graph, are uniquely identified by their adjacency and Laplacian spectra, extending spectral characterization to these graphs.

Contribution

It establishes that connected multicone graphs and their complements are determined by their spectra, generalizing spectral determination results for friendship graphs.

Findings

Connected multicone graphs are determined by their adjacency spectra.

Laplacian spectra also uniquely identify these graphs.

Complements of multicone graphs (except when s=2) are spectrally determined.

Abstract

In this study we investigate the spectra of the family of connected multicone graphs. A multicone graph is defined to be the join of a clique and a regular graph. Let $ r $, $ t $ and $ s $ be natural numbers, and let $ K_r $ denote a complete graph on $ r $ vertices. It is proved that connected multicone graphs $ K_r\bigtriangledown sK_t $, a natural generalization of friendship graphs, are determined by their adjacency spectra as well as their Laplacian spectra. Also, we show that the complement of multicone graphs $ K_r\bigtriangledown sK_t $ are determined by their adjacency spectra, where $ s\neq 2 $.